where \(C_{\mathcal {G}\mathcal {G}}\)  is the empirical covariance of the
ensemble of quantities of interest from model runs, and \(C_{\mathbf {\theta }\mathcal {G}}\)  is
the empirical cross-covariance of the ensemble of parameters and the ensemble of quantities of interest. The covariance matrix of the distribution of differences between realizations of \(\mathbf {y}\)  and \(\mathcal {G}_T(\cdot )\)  is \(\Gamma +\Sigma \) , which is approximated empirically in the update by \(\Gamma +C_{\mathcal {G}\mathcal {G}}^{(n)}\) .
When ensemble methods are used as approximate samplers [1]}, [2]}, additional independent noise is added to \(\mathbf {y}\)  at each iteration
and for every ensemble member; however, because
we are solving an optimization problem within this calibration phase, such noise is not added here.
